MathDB

12

Part of 2023 MMATHS

Problems(2)

MMATHS 2023 Individual Problem 12: Hyperbola through 5 points

Source:

9/24/2024
Let ABCABC be a triangle with incenter I.I. The incircle ω\omega of ABCABC is tangent to sides BC,CA,BC, CA, and ABAB at points D,E,D, E, and F,F, respectively. Let D′D' be the reflection of DD over I.I. Let PP be a point on ω\omega such that ∠ADP=90∘.\angle{ADP}=90^\circ. H\mathcal{H} is a hyperbola passing through D′,E,F,I,D', E, F, I, and P.P. Given that ∠BAD=45∘\angle{BAD}=45^\circ and ∠CAD=30∘,\angle{CAD}=30^\circ, the acute angle between the asymptotes of H\mathcal{H} can be expressed as (mn)∘,\left(\tfrac{m}{n}\right)^\circ, where mm and nn are relatively prime positive integers. Find m+n.m+n.
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MMATHS 2023 Team Problem 12: An incenter and excenter configuration

Source:

9/24/2024
Let ABCABC be a triangle with incenter I,I, circumcenter O,O, and AA-excenter JA.J_A. The incircle of △ABC\triangle{ABC} touches side BCBC at a point D.D. Lines OIOI and JADJ_AD meet at a point K.K. Line AKAK meets the circumcircle of △ABC\triangle{ABC} again at a point L≠A.L \neq A. If BD=11,CD=5,BD=11, CD=5, and AO=10,AO=10, the length of DLDL can be expressed as mpn,\tfrac{m\sqrt{p}}{n}, where m,n,pm,n,p are positive integers, mm and nn are relatively prime, and pp is not divisible by the square of any prime. Find m+n+p.m+n+p.
YaleMMATHS